If P is a topological property, then a space (X, τ) is said to be minimal P (respectively, maximal) if (X,τ) has property P but no topology on X which is strictly smaller (respectively, strictly larger ) than τ has P.

A topological space is called KC space if every compact subset is closed.

A topological space is called strongly KC space if every countably compact subset is closed.

I know (strongly) KC – space is topological property.

I want to know:

Is there a example of a minimal KC but not minimal strongly KC – space?

It is a fact that an infinite minimal strongly KC space has a trivial convergent sequence, but it does not hold for minimal KC space. I want an easy example to show it.

$\begingroup$Dear fatemeh: I answered your previous question about KC spaces, and apparently you didn´t even read it or didn´t bother to either accept it or comment on the answer.$\endgroup$
– Ramiro de la VegaJul 26 '13 at 22:22

2

$\begingroup$A small suggestion: I think the several .......... appearing in the question are unnecessary. Perhaps you just want to start a new paragraph?$\endgroup$
– Ricardo AndradeJul 26 '13 at 22:47